In this article, we are concerned with the Dirichlet problem of the stationary von Neumann-Landau wave equation：{（-△x＋△y）φ（x,y）=0,x,y∈Ωφ｜δΩxδΩ=fwhere Ω is a bounded domain in R^n. By introducing anti...In this article, we are concerned with the Dirichlet problem of the stationary von Neumann-Landau wave equation：{（-△x＋△y）φ（x,y）=0,x,y∈Ωφ｜δΩxδΩ=fwhere Ω is a bounded domain in R^n. By introducing anti-inner product spaces, we show the existence and uniqueness of the generalized solution for the above Dirichlet problem by functional-analytic methods.展开更多
In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 bou...In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.展开更多
In this paper, we study the Holder regularity of weak solutions to the Dirichlet problem associated with the regional fractional Laplacian (-△)^(α)Ω on a bounded open set Ω ■R(N ≥ 2) with C^((1,1)) boundary ■Ω...In this paper, we study the Holder regularity of weak solutions to the Dirichlet problem associated with the regional fractional Laplacian (-△)^(α)Ω on a bounded open set Ω ■R(N ≥ 2) with C^((1,1)) boundary ■Ω. We prove that when f ∈ L^(p)(Ω), and g ∈ C(Ω), the following problem (-△)^(α)Ωu = f in Ω, u = g on ■Ω, admits a unique weak solution u ∈ W^((α,2))(Ω) ∩ C(Ω),where p >N/2-2α and 1/2< α < 1. To solve this problem, we consider it into two special cases, i.e.,g ≡ 0 on ■Ω and f ≡ 0 in Ω. Finally, taking into account the preceding two cases, the general conclusion is drawn.展开更多
In this paper, we study the existence of nontrivial radial convex solutions of a singular Dirichlet problem involving the mean curvature operator in Minkowski space. The proof is based on a well-known fixed point theo...In this paper, we study the existence of nontrivial radial convex solutions of a singular Dirichlet problem involving the mean curvature operator in Minkowski space. The proof is based on a well-known fixed point theorem in cones. We deal with more general nonlinear term than those in the literature.展开更多
In this article, a class of Dirichlet problem with Lp boundary data for poly-harmonic function in the upper half plane is mainly investigated. By introducing a sequence of kernel functions called higher order Poisson ...In this article, a class of Dirichlet problem with Lp boundary data for poly-harmonic function in the upper half plane is mainly investigated. By introducing a sequence of kernel functions called higher order Poisson kernels and a hierarchy of integral operators called higher order Pompeiu operators, we obtain a main result on integral representation solution as well as the uniqueness of the polyharmonic Dirichlet problem under a certain estimate.展开更多
We study the Dirichlet problem of the n-dimensional complex Monge-Ampere equation det（uij） = F/｜z｜2a, where 0 〈 a 〈 n. This equation comes from La Nave-Tian＇s continuity approach to the Analytic Minimal Model P...We study the Dirichlet problem of the n-dimensional complex Monge-Ampere equation det（uij） = F/｜z｜2a, where 0 〈 a 〈 n. This equation comes from La Nave-Tian＇s continuity approach to the Analytic Minimal Model Program.展开更多
In this paper, we study the Dirichlet problems for the following quasilinear secondorder sub-elliptic equation, sum from i,j=1 to m(X~*(A(x,u)Xu)+sum from j=1 to m(B(x,u)Xu+C(x,u)=0 in Ω, u=φ on Ω,where X={X, …, ...In this paper, we study the Dirichlet problems for the following quasilinear secondorder sub-elliptic equation, sum from i,j=1 to m(X~*(A(x,u)Xu)+sum from j=1 to m(B(x,u)Xu+C(x,u)=0 in Ω, u=φ on Ω,where X={X, …, X} is a system of real smooth vector fields which satisfies the Hrmander’scondition, A(i,j), B, C∈C~∞(■×R) and (A(x, z)) is a positive definite matris. We have provedthe existence and the maximal regularity of solutions in the "non-isotropic" Hlder space associatedwith the system of vector fields X.展开更多
In this paper, the author studies the regularity of solutions to the Dirichlet problem forequation Lu = f, where L is a second order degenerate elliptic operator in divergence form inΩ, a bounded open subset of Rn (n...In this paper, the author studies the regularity of solutions to the Dirichlet problem forequation Lu = f, where L is a second order degenerate elliptic operator in divergence form inΩ, a bounded open subset of Rn (n ≥ 3).展开更多
Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equatio...Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain of the second type is discussed by using the analytic method in this paper. Firstly, the complex Monge-Ampère equation is reduced to a nonlinear second-order ordinary differential equation (ODE) by using quite different method. Secondly, the solution of the Dirichlet problem is given in semi-explicit formula, and under a special case the exact solution is obtained. These results may be helpful for the numerical method of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain.展开更多
We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation：where ω 〉 0, a（x） is a continuous function satisfying 0 〈 a（x） 〈 1 for...We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation：where ω 〉 0, a（x） is a continuous function satisfying 0 〈 a（x） 〈 1 for x ∈Ω, Ω is a bounded smooth domain in R^N. We will see that the profile of a minimal positive boundary blow-up solution of the equation shares some similarities to the profile of a positive minimizer solution of the equation with homogeneous Dirichlet boundary condition.展开更多
Taking hm as the mesh width of a curved edge Гm （m = 1, ..., d ） of polygons and using quadrature rules for weakly singular integrals, this paper presents mechanical quadrature methods for solving BIES of the first...Taking hm as the mesh width of a curved edge Гm （m = 1, ..., d ） of polygons and using quadrature rules for weakly singular integrals, this paper presents mechanical quadrature methods for solving BIES of the first kind of plane elasticity Dirichlet problems on curved polygons, which possess high accuracy O（h0^3） and low computing complexities. Since multivariate asymptotic expansions of approximate errors with power hi^3 （i = 1, 2, ..., d） are shown, by means of the splitting extrapolations high precision approximations and a posteriori estimate are obtained.展开更多
In this paper,we consider the exterior Dirichlet problem of Hessian equationsσk(λ(D^(2)u))=g(x)with g being a perturbation of a general positive function at infinity.The existence of the viscosity solutions with gen...In this paper,we consider the exterior Dirichlet problem of Hessian equationsσk(λ(D^(2)u))=g(x)with g being a perturbation of a general positive function at infinity.The existence of the viscosity solutions with generalized asymptotic behavior at infinity is established by the Perron’s method which extends the previous results for Hessian equations.By the solutions of Bernoulli ordinary differential equations,the viscosity subsolutions and supersolutions are constructed.展开更多
ASSUME that ξ = （ξ<sub>t</sub>, Ⅱ<sub>x</sub>） is a right Markov process in R<sup>d</sup>. Let φ（x, z） = a（x）z+b（x）z<sup>2</sup>+integral from n=0 to ∞ (e&l...ASSUME that ξ = （ξ<sub>t</sub>, Ⅱ<sub>x</sub>） is a right Markov process in R<sup>d</sup>. Let φ（x, z） = a（x）z+b（x）z<sup>2</sup>+integral from n=0 to ∞ (e<sup>-uz</sup>-1+uz)n<sup>x</sup>（du）, x∈ R<sup>d</sup>, z∈R<sup>+</sup>. （1）Consider the following Dirichlet problem:展开更多
In this paper, we study a nonlinear Dirichlet problem on a smooth bounded domain, in which the nonlinear term is asymptotically linear, not superlinear, at infinity and sublinear near the origin. By using Mountain Pas...In this paper, we study a nonlinear Dirichlet problem on a smooth bounded domain, in which the nonlinear term is asymptotically linear, not superlinear, at infinity and sublinear near the origin. By using Mountain Pass Theorem, we prove that there exist at least two positive solutions under suitable assumptions on the nonlinearity展开更多
Given a Markov process satisfying certain general type conditions,whose paths are notassumed to be continuous. Let D by an open subset of the state space E. Any bounded function defined on thecomplement of D extends t...Given a Markov process satisfying certain general type conditions,whose paths are notassumed to be continuous. Let D by an open subset of the state space E. Any bounded function defined on thecomplement of D extends to be a function on E (?)uch that it is harmonic in D and satisfies the Dirichletboundary condition at any regular boundary point of D. The relation between harmonic functions and theebaracteristic operator of the given process is discussed.展开更多
This paper is devoted to the study of second order nonlinear difference equations. A Nonlocal Perturbation of a Dirichlet Boundary Value Problem is considered. An exhaustive study of the related Green＇s function to t...This paper is devoted to the study of second order nonlinear difference equations. A Nonlocal Perturbation of a Dirichlet Boundary Value Problem is considered. An exhaustive study of the related Green＇s function to the linear part is done. The exact expression of the function is given, moreover the range of parameter for which it has constant sign is obtained. Using this, some existence results for the nonlinear problem are deduced from monotone iterative techniques, the classical Krasnoselski fixed point theorem or by application of recent fixed point theorems that combine both theories.展开更多
Using invariant sets of descending flow and variational methods, we establish some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for second-order nonlinea...Using invariant sets of descending flow and variational methods, we establish some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for second-order nonlinear difference equations with Dirichlet boundary value problem. Some results in the literature are improved.展开更多
Let L be a second-order linear elliptic operator with complex coefficients. It is shown that if the L^p Dirichlet problem for the elliptic system L(u) = 0 in a fixed Lipschitz domain Ω in Rd is solvable for some 1 &l...Let L be a second-order linear elliptic operator with complex coefficients. It is shown that if the L^p Dirichlet problem for the elliptic system L(u) = 0 in a fixed Lipschitz domain Ω in Rd is solvable for some 1 < p = p_0 <2(d-1)/(d-2), then it is solvable for all p satisfying ■ The proof is based on a real-variable argument. It only requires that local solutions of L(u) = 0 satisfy a boundary Cacciopoli inequality.展开更多
In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equ...In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and quasi-derivatives.展开更多
基金Supported partially by the National Natural Science Foundation of China(10775175)
文摘In this article, we are concerned with the Dirichlet problem of the stationary von Neumann-Landau wave equation：{（-△x＋△y）φ（x,y）=0,x,y∈Ωφ｜δΩxδΩ=fwhere Ω is a bounded domain in R^n. By introducing anti-inner product spaces, we show the existence and uniqueness of the generalized solution for the above Dirichlet problem by functional-analytic methods.
基金supported by the National NaturalScience Foundation of China(11971069 and 12126307)。
文摘In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.
基金Supported by the Natural Science Foundation of Hebei Province (Grant No. A2018210018)the Science and Technology Research Program of Higher Educational in Hebei Province (Grant No. ZD2019047)。
文摘In this paper, we study the Holder regularity of weak solutions to the Dirichlet problem associated with the regional fractional Laplacian (-△)^(α)Ω on a bounded open set Ω ■R(N ≥ 2) with C^((1,1)) boundary ■Ω. We prove that when f ∈ L^(p)(Ω), and g ∈ C(Ω), the following problem (-△)^(α)Ωu = f in Ω, u = g on ■Ω, admits a unique weak solution u ∈ W^((α,2))(Ω) ∩ C(Ω),where p >N/2-2α and 1/2< α < 1. To solve this problem, we consider it into two special cases, i.e.,g ≡ 0 on ■Ω and f ≡ 0 in Ω. Finally, taking into account the preceding two cases, the general conclusion is drawn.
基金supported by the Key Program of Scientific Research Fund for Young Teachers of AUST(QN2018109)the National Natural Science Foundation of China(11801008)+1 种基金supported by the Fundamental Research Funds for the Central Universities(2017B715X14)the Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX17_0508)
文摘In this paper, we study the existence of nontrivial radial convex solutions of a singular Dirichlet problem involving the mean curvature operator in Minkowski space. The proof is based on a well-known fixed point theorem in cones. We deal with more general nonlinear term than those in the literature.
文摘In this article, a class of Dirichlet problem with Lp boundary data for poly-harmonic function in the upper half plane is mainly investigated. By introducing a sequence of kernel functions called higher order Poisson kernels and a hierarchy of integral operators called higher order Pompeiu operators, we obtain a main result on integral representation solution as well as the uniqueness of the polyharmonic Dirichlet problem under a certain estimate.
基金supported by NSFC(Grant No.11331001)supported by NSFC(Grant No.11501285)
文摘We study the Dirichlet problem of the n-dimensional complex Monge-Ampere equation det（uij） = F/｜z｜2a, where 0 〈 a 〈 n. This equation comes from La Nave-Tian＇s continuity approach to the Analytic Minimal Model Program.
文摘In this paper, we study the Dirichlet problems for the following quasilinear secondorder sub-elliptic equation, sum from i,j=1 to m(X~*(A(x,u)Xu)+sum from j=1 to m(B(x,u)Xu+C(x,u)=0 in Ω, u=φ on Ω,where X={X, …, X} is a system of real smooth vector fields which satisfies the Hrmander’scondition, A(i,j), B, C∈C~∞(■×R) and (A(x, z)) is a positive definite matris. We have provedthe existence and the maximal regularity of solutions in the "non-isotropic" Hlder space associatedwith the system of vector fields X.
文摘In this paper, the author studies the regularity of solutions to the Dirichlet problem forequation Lu = f, where L is a second order degenerate elliptic operator in divergence form inΩ, a bounded open subset of Rn (n ≥ 3).
基金supported by the Research Foundation of Beijing Government(Grant No.YB20081002802)National Natural Science Foundation of China(Grant No.10771144)
文摘Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain of the second type is discussed by using the analytic method in this paper. Firstly, the complex Monge-Ampère equation is reduced to a nonlinear second-order ordinary differential equation (ODE) by using quite different method. Secondly, the solution of the Dirichlet problem is given in semi-explicit formula, and under a special case the exact solution is obtained. These results may be helpful for the numerical method of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain.
基金Supported by National Natural Science Foundation of China （Grant No. 10871060）
文摘We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation：where ω 〉 0, a（x） is a continuous function satisfying 0 〈 a（x） 〈 1 for x ∈Ω, Ω is a bounded smooth domain in R^N. We will see that the profile of a minimal positive boundary blow-up solution of the equation shares some similarities to the profile of a positive minimizer solution of the equation with homogeneous Dirichlet boundary condition.
基金Supported by Natioal Science Foundation of China （10171073）.
文摘Taking hm as the mesh width of a curved edge Гm （m = 1, ..., d ） of polygons and using quadrature rules for weakly singular integrals, this paper presents mechanical quadrature methods for solving BIES of the first kind of plane elasticity Dirichlet problems on curved polygons, which possess high accuracy O（h0^3） and low computing complexities. Since multivariate asymptotic expansions of approximate errors with power hi^3 （i = 1, 2, ..., d） are shown, by means of the splitting extrapolations high precision approximations and a posteriori estimate are obtained.
基金supported in part by the Natural Science Foundation of Tianjin City of China(Grant No.19JCQNJC14700).
文摘In this paper,we consider the exterior Dirichlet problem of Hessian equationsσk(λ(D^(2)u))=g(x)with g being a perturbation of a general positive function at infinity.The existence of the viscosity solutions with generalized asymptotic behavior at infinity is established by the Perron’s method which extends the previous results for Hessian equations.By the solutions of Bernoulli ordinary differential equations,the viscosity subsolutions and supersolutions are constructed.
文摘ASSUME that ξ = （ξ<sub>t</sub>, Ⅱ<sub>x</sub>） is a right Markov process in R<sup>d</sup>. Let φ（x, z） = a（x）z+b（x）z<sup>2</sup>+integral from n=0 to ∞ (e<sup>-uz</sup>-1+uz)n<sup>x</sup>（du）, x∈ R<sup>d</sup>, z∈R<sup>+</sup>. （1）Consider the following Dirichlet problem:
基金the National Natural Science Foundation of China.
文摘In this paper, we study a nonlinear Dirichlet problem on a smooth bounded domain, in which the nonlinear term is asymptotically linear, not superlinear, at infinity and sublinear near the origin. By using Mountain Pass Theorem, we prove that there exist at least two positive solutions under suitable assumptions on the nonlinearity
文摘Given a Markov process satisfying certain general type conditions,whose paths are notassumed to be continuous. Let D by an open subset of the state space E. Any bounded function defined on thecomplement of D extends to be a function on E (?)uch that it is harmonic in D and satisfies the Dirichletboundary condition at any regular boundary point of D. The relation between harmonic functions and theebaracteristic operator of the given process is discussed.
基金partially supported by Ministerio de Educación y Ciencia,Spain,and FEDER,Projects MTM2013-43014-P and MTM 2016-75140-P
文摘This paper is devoted to the study of second order nonlinear difference equations. A Nonlocal Perturbation of a Dirichlet Boundary Value Problem is considered. An exhaustive study of the related Green＇s function to the linear part is done. The exact expression of the function is given, moreover the range of parameter for which it has constant sign is obtained. Using this, some existence results for the nonlinear problem are deduced from monotone iterative techniques, the classical Krasnoselski fixed point theorem or by application of recent fixed point theorems that combine both theories.
文摘Using invariant sets of descending flow and variational methods, we establish some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for second-order nonlinear difference equations with Dirichlet boundary value problem. Some results in the literature are improved.
文摘Let L be a second-order linear elliptic operator with complex coefficients. It is shown that if the L^p Dirichlet problem for the elliptic system L(u) = 0 in a fixed Lipschitz domain Ω in Rd is solvable for some 1 < p = p_0 <2(d-1)/(d-2), then it is solvable for all p satisfying ■ The proof is based on a real-variable argument. It only requires that local solutions of L(u) = 0 satisfy a boundary Cacciopoli inequality.
基金supported by National Natural Science Foundation of China(Grant No.11631004)Science and Technology Commission of Shanghai Municipality(Grant No.14XD1400400)
文摘In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and quasi-derivatives.
